Introduction To Research In the Classroom

Answers To Frequently Asked Questions

This is an abridged version of a longer FAQ list compiled by Joshua Abrams for Making Mathematics,
an NFS - funded project based at Center for Mathematics Eductaion of Education Development Center (http://www2.edc.org/cme/).

Mathematics research is the long-term, open-ended exploration of a set
of related mathematics questions whose answers connect to and build upon
each other. Problems are open-ended because students continually come
up with new questions to ask based on their observations. Additional characteristics
of student research include:

Students develop questions, approaches, and results, that are, at
least for them, original products.

Students use the same general methods used by research mathematicians.
They work through cycles of data-gathering, visualization, abstraction,
conjecturing, and proof.

Students communicate mathematically: describing their thinking, writing
definitions and conjectures, using symbols, justifying their conclusions,
and reading mathematics.

When the research involves a class or other group, the students become
a community of mathematicians sharing and building on each others
questions, conjectures, and theorems.

How do students benefit from doing mathematics
research?

Mathematics research influences student learning in a number of ways:

Research provides students with an understanding of what it means
to do mathematics and of mathematics as a living, growing field.

Writing mathematics and problem-solving become central to students
learning.

Students develop mastery of mathematics topics.
Philosopher and educator John Dewey claimed that we
dont learn the basics by studying the basics but by engaging in
rich activities which require them. Research experiences require the
repeated application of technical skills in the service of looking for
patterns and testing conjectures. It is this repetition, in the context of
motivating and meaningful problems, that leads to greater understanding
and retention of mathematics skills. During an investigation, students
make connections between ideas that further enhance retention.

Students develop their own mathematical aesthetic as they practice
making choices about which aspects of a problem to investigate.

Students develop both confidence as mathematical thinkers and enthusiasm
to do more mathematics. The creativity, problem-solving, surprises,
and accomplishments that are part of research help to answer students
questions about the value of studying mathematics. They are studying
new methods so that they can answer their own questions. They are learning
in order to do work that they care about at that moment (and not for
a test or some far-off future task).

Doing research is challenging and can be frustrating. Students
commitment to persistence and tolerance for frustration grow as they
are supported, encouraged, and given repeated opportunity to think about
and succeed with problems over days and weeks.

Students learn to distinguish between different levels of evidence
and to be skeptical in the face of anecdotal evidence. The habit of
looking for counterexamples to claims is a core skill for critical thinkers
in all aspects of life.

For which students is research appropriate?

This question is usually more bluntly framed as "Can kids really do this?!"
The experience of teachers in all types of school settings is that all
children can successfully engage in mathematics research. In Making Mathematics,
a recently completed project based at Education Development Center,
teachers undertook research with urban, rural, and suburban students
from grades 4 through 12. They guided at-risk, honors, and English
as a Second Language (ESL) classes through projects lasting from a few
weeks up to a year. Students in math clubs, individual students, and home-schooled
students carried out successful investigations. One teacher
first introduced research to her honors seventh graders. Once she was
confident in her own experience, she tried the same project with two low-tracked
eighth-grade sections. The quality of the questions, experimenting, reasoning,
and writing was excellent in all three sections and indistinguishable
between the honors and non-honors students. Research drew upon a richer
array of student abilities than were assessed for tracking purposes.

Research can thrive in a heterogeneous class of students if you pick
a project that does not require a lot of background to get started but
which also inspires sophisticated questions. Students will pose problems
at a level that is both challenging and appropriate for them.

How can I get my feet wet with research?

Making Mathematics teachers have been most comfortable trying research
for the first time with one of their "stronger than average" sections.
Some teachers have begun work with one or more interested students as
part of a mathematics club or independent seminar. The purpose of these
first excursions has been for the students to become familiar with the
research process and for the teacher to see how students respond to lengthy,
open-ended problem-solving.

You should commit at least three consecutive class periods at the start
of a first investigation in order to maintain the momentum of the experience.
You want students to appreciate that the questions are not typical quick
exercises, so it is important that they get to wade into the work. Interruptions
also make it harder for them to maintain a line of thinking. After the
initial burst, you can sustain a project through weekly discussions of
work done at home. If a problem is working well, do not be afraid to let
kids pursue it for a long period of time.

What kind of support will I need?

Many teachers independently introduce research into a class. Your work
will have greater impact on students if they encounter research in all
of their mathematics classes. Both for that reason and in order to feel
less isolated as you experiment, it is helpful to recruit one or more
colleagues to try out research along with you. Share ideas and observations
and even visit each others classes on days when the students are
doing research. Talk with your department head or supervisor to garner
support for your efforts.

Mathematicians in the Focus on Mathematics program would all be eager to serve as a mentor
for you and your students. The rest of this paragraph describes how you might search for a mentor
if independently of FOM. If you want an advisor for yourself or an outside audience for the work
that your students do, you can contact the mathematics or mathematics
education department at a local college and ask if any of the professors
would be willing to serve as a mentor (either via email, phone, or in
person) for you and your class. We have also found good mentors contacting
corporations that employ scientists and mathematicians. Your mentor may
just communicate with you or she may be willing to read updates or reports
from the students and provide responses. You should make these exchanges
via your email accountparental consent is required by law
for direct internet communication. Be sure to let any prospective mentor
know what your goals and expectations are for the students and for their
involvement.

Mentors can help in a number of ways. They can:

Help pick an area of inquiry and establish goals.

Make suggestions about next steps (e.g., when to apply a research
technique or content area, whether to redirect a classs
efforts).

Ask clarifying questions about the students mathematical
statements.

Help students learn how to prove their claims.

Study and reflect upon student work with you.

Provide an authentic outside audience for student efforts.

Provide emotional support such as encouragement, perspective, and
advice.

Identify resources.

What do I need to do before I begin?

If you have never done any mathematics research yourself, it is time
to join in! Your FOM mathematicians will help you and your colleagues, pick a project,
and start your work looking for patterns, trying to state clear conjectures,
searching for proofs or disproofs, and studying new, related problems.
Many Making Mathematics teachers have found the summer a good
time for professional growth via a research project.

Decide what your goals for your initial foray are (e.g., each group
will be responsible for posing and resolving one question or students
will each have their own conjecture) and how much time you plan to spend
on the project.

Pick a project topic.

Since research is unfamiliar to many parents, you
may want to anticipate any questions that will arise by discussing your
plans ahead of time. You can send a letter home to parents that helps them
to understand what you will be doing and why.

What might a research sequence within a class look like?

The teaching notes accompanying the Making Mathematics
projects (http://www2.edc.org/makingmath/) can serve as models that you can adapt to other projects. As
noted earlier, it is best if you can introduce research
with a burst that permits a coherent presentation of the research process
before separating discussions with several days of non-research studies.

Once research is underway, each student or group of students may work
on different, but related, questions. During whole-class discussion, classmates
should describe the different problems that they are exploring. Students
should report back on their progress (new questions, conjectures, proofs,
etc.) periodically.

At the end of a class session devoted to research, each group should
give themselves a homework assignment in a logbooks.
You can check these recorded tasks to make sure that the assignments were
meaningful and check the subsequent entry in the logbook to make sure
that the student made reasonable progress with the tasks. Typical homework
challenges include:

Extend a pattern, generate more data.

Try to prove a particular conjecture.

Test a bunch of conjectures with different cases to see whether counterexamples
can be found.

Try to find a formula or rule for a pattern.

Identify and learn about areas of mathematics that might be helpful
to the investigation.

Read about related problems and how they were solved.

Pose extensions of the project.

Students can think about where they are in the research
process (see below for one model for the process) in order to decide what step to attempt next.
Their work should
have some narrative explanations ("I did this because "). Students
can work on their homework for a few days, but groups will also need regular
class time to catch up on each others thinking, to work together,
and to then coordinate next steps before their next stretch of independent
work.

Although some projects, such as the one in Making Mathematics include teaching notes of some kind that
suggest what to do on the first day, the second day, and so forth, you
will need to pace the phases of a particular investigation according to
the length of your class periods and the timing of a given classs
particular questions and discoveries. Here are some other decisions that
you should be alert to as work proceeds:

Students will naturally exhibit important research skills such as
posing a conjecture, organizing data in an effective manner, or inventing
a new definition. When this happens, you want to identify the skill
and discuss its importance to research. For example, a student might
note the existence of a counter-example to a classmates conjecture.
If students do not already know about counter-examples, you could stop
to highlight the contribution and do a side lesson on examples
(generating test cases, remaining skeptical in the face of confirming
examples, extreme and degenerate cases, and counter-examples).

Issues will arise unpredictably and student comments may simultaneously
pull a class in a number of possible directions. For example, one student
may invoke a counter-example for the first time and another might pose
two new questions to explore. Where do you head first? You should try
to strike a balance between developing a formal understanding of research
skills and allowing the research process to unfold without too many
interruptions. It is always good to at least give a name to a new research
skill or problem-solving strategy when a student demonstrates it. You
can then return to a discussion of the considerations associated with
that habit in depth at a later time.

As an investigation continues, the difficulty of generating further
examples may become an impediment to further progress. You should help
the students decide whether carrying out the steps needed to find new
examples is still itself illuminating or whether just the data gleaned
from the examples is what they need. If the latter is the case and there
is a way to use technology to speed up the work, it might be worth taking
the time to teach the class or a particular group how to use the appropriate
tool. For example, they might benefit from assistance doing symbolic
manipulations using a computer algebra system (CAS), geometric constructions
using a dynamic geometry program, or finding numeric examples using
a spreadsheet.

As a class works thorough its early research experiences, be sure to
document for them as much of their work as possible. Posters listing the
students conjectures, questions, and theorems help students grasp
the cyclical nature of the research process. They see how their different
questions connect and build upon each other and learn which research methods
are most helpful at which stages of an investigation. After these beginning
projects, students are ready to work more independently and should be
encouraged to pose their own questions for research.

How does a research project
end?

A project can end when a student or group has resolved some central question.
Often, there are many questions and, after good progress with some of
them, students enthusiasm for the others may wane. You may have
established certain goals for students: to create a proof, to generate
a few clear conjectures, to pose a new problem and make progress with
it. Each of these possibilities is a reasonable time for work on a project
to end. Students can come to a satisfying sense of closure even with a
project that leaves many unanswered questions. That feeling can be enhanced
if they write a final report that summarizes their main questions and
work and that concludes with a list of possible extensions worth exploring.
The FOM mathematician can help you with ideas about formal write-ups
for students who have engaged research project.

How will doing
research affect my workload?

Ultimately, research is no more demanding on your time than teaching
that is more traditional. In some cases, it shifts the balance so that
you spend less time preparing lessons and more time responding to student
work. If you have not taught research before, there will be an initial
need to think through the different issues that will arise in class. This
work will prepare you to take advantage of any "teachable moments" (student
comments that can lead the class to new understandings). The Making Mathematicsteacher
handbook is a valuable resource as you develop experience doing research
with students.

One strategy for managing the demands of teaching research is to keep
good notes on your observations during class. Thorough ongoing documentation
will facilitate the comments that you need to make when you collect work
because you will have a good sense of the entire research process that
an individual or group has gone through. The more often you can read and
respond to students entries in a their logbooks, the better, but
you do not have to collect everyones work all at once. You can sample
a few each night. Lastly, having each group submit a single final report
reduces the number of papers that you need to study to a manageable number.

How can I balance the development of research
skills with the need to cover specific mathematics topics?

Teacher: Since I have a curriculum I have to cover, which really takes
the whole school year, how do I cover that curriculum while implementing
those problems that deepen the understanding of the required curriculum
and is there some way to perhaps replace chunks of my text with such
problems? I find balancing what I WANT to do and what I MUST do very
difficult.

Mentor: I appreciate your frustration about the tension between covering
technical content and giving your students the opportunity to learn
about the process of doing mathematics. There is no question that teachers
are being asked to whiz through too many topics. I try to remind teachers
of what they already know: when we go too quickly, the material is not
mastered well and so we are not being efficient.

The above exchange between a Making Mathematics teacher and her mentor
is typical of the most common and emotional question with which teachers
interested in research have grappled. Many have expressed stress at feeling
trapped by competing demands. In some cases, the answer is simple: if
there is a major state test next week and you need to cover five topics,
it is definitely a bad time to start research. But, if you are months
away and you consider how often students forget what they have studied,
now is a good time to introduce your students to mathematics investigations.

"There is, of course, a cost to having the students engage more deeply
in the mathematics: one "covers" less. However, when the payoffs include
much deeper understanding, much longer retention of the content, enthusiasm,
and the fact that the students get a much better sense of the mathematical
enterprise, the price in (ostensible) coverage is a small one to pay."
 Mathematics Educator Alan Schoenfeld.

The content
versus research question reflects a false dichotomy. We know how fruitless
it is to teach disconnected topics. If you do not use knowledge in active
ways that allow you to make meaning of what you have learned, you do not
retain that learning. Why do students seem to forget so much of what they
study? Sometimes, they still have the skills but are only able to apply
them when prompted (e.g., "I am doing a chapter four problem" or "I was
told to use triangle trigonometry techniques"). Sometimes, the learning
experience was not memorable (consider what you have remembered and forgotten
from high school and try to identify why). The more research work becomes
a strand throughout a course and a schools curriculum, the better
the interconnections between, and mastery of, technical content will be.

The NCTM Standards include many important goals (e.g., being able to
conjecture, show persistence in problem solving, develop mathematical
models, etc.) that we are supposed to "cover" that do not fit well in
the framework of timed tests.

So, how do we combine research and technical content goals and what are
some of the challenges that we face in our efforts? We can choose a research
problem that will reinforce technical skills that a class has already
studied. Alternatively, we can pick a problem that will introduce our
students to and help them develop an understanding of a new topic. For
example, we could use the Game of Set research project
in place of or after a textbook introduction on combinatorics.

One problem that arises when using a research experience as a way to
develop or reinforce a particular technical skill is that students
questions and methods may not head in the direction that you expected.
If you tell students to use a particular technique,
then you short-circuit the research process. You are also risking turning
the effort into a planned discovery activity, which usually lacks the
motivational and intellectual power of true research.

You can address this problem in a few ways. A careful choice of project
or framing of the question can often make certain skills inevitable. For
example, a high school class proving theorems about Pythagorean
Triples would be hard pressed to avoid using algebraic expressions
or thinking about factors. You can also add your own questions to the
classs list. This makes you a participant in the process and assures
that the class will spend some time on the issues that you want considered.
Alternatively, you can let the students work take them where it
will knowing that some other important area of mathematics is being developed
or reinforced that you will not have to spend as much time on in the future.
Then, after the research is over, you can return to the topic that you
originally had in mind.

When students do get to follow their own intellectual muse, they are
more likely to experience a wide range of mathematics topics. For example,
in a class of fifth graders working on the Connect the Dots project, one
student asked what would happen if each jump was chosen randomly. The
shapes were no longer as attractive, but the question of whether they
would ever close led to the idea of expected value. An independent research
project on randomness in DNA led a student to study matrices and Markov
processes. Students will teach themselves a chapter of content from a
textbook if they think it will help them on a task about which they care.